Đáp án:
$\begin{array}{l}
Đặt:\dfrac{a}{b} = \dfrac{b}{c} = \dfrac{c}{a} = k\\
\Rightarrow \left\{ \begin{array}{l}
a = b.k\\
b = c.k\\
c = a.k
\end{array} \right.\\
\Rightarrow a = c.k.k = a.k.k.k = a.{k^3}\\
\Rightarrow {k^3} = \dfrac{a}{a} = 1\\
\Rightarrow k = 1\\
\Rightarrow \dfrac{a}{b} = \dfrac{b}{c} = \dfrac{c}{a} = 1\\
\Rightarrow a = b = c\\
M = \dfrac{{{a^{2019}} + {b^{2019}} + {c^{2019}}}}{{{a^{672}}.{b^{673}}.{c^{674}}}}\\
= \dfrac{{{a^{2019}} + {a^{2019}} + {a^{2019}}}}{{{a^{672}}.{a^{673}}.{a^{674}}}}\\
= \dfrac{{3.{a^{2019}}}}{{{a^{672 + 673 + 674}}}}\\
= \dfrac{{3.{a^{2019}}}}{{{a^{2019}}}}\\
= 3
\end{array}$
Vậy M=3
